On Substitutivity

On Substitutivity

The idea of substituting one expression for another has played a key role in logical and semantic studies. In particular, the idea of substituting terms with the same reference has featured prominently: can this always be done without changing the truth-value of the sentence in which the terms occur? Is such substitution ever truth-value disruptive? The consensus has been that it can be, for example when substituting into belief contexts. Thus, the convention has arisen of calling some contexts “referentially transparent” and some “referentially opaque”. The distinction has been thought to be binary, not divisible into finer distinctions. I think this is a mistake; it misses important differences. There are at least four kinds of case to consider, each with distinctive properties and varying explanations. Specifically, there are three types of opacity, which deserve different names; at least two of them exhibit their own kind of transparency—they are not fully opaque. Let’s accept, for the moment, that there are no other kinds of transparency wider than the usual kind; we can then ask whether there are degrees or grades of opacity, i.e., departures from full transparency.

First, consider modal, causal, and explanatory contexts: just how opaque are these? Suppose I say “The king of England is necessarily a king”, giving the modal operator wide scope: that is clearly true, analytically so. But it is not true if we substitute “Charles Windsor” for “the king of England”—hence the context is not referentially transparent. Can we substitute any other expression for the description and obtain a truth? Yes, if we restrict the substituted terms to those that make reference to kinghood: “the male monarch”, “the male head of the church of England”, “the male hereditary ruler of an independent state”, etc. The germane consideration is that the property of being a king necessarily implies being a king (or a monarch more generally); it doesn’t matter whether anyone knows or believes this to be so—it is not dependent on how kings are thought of or mentally represented. It is a modal fact that obtains independently of how anyone thinks (compare “the successor of 2 is a number”). So, this context is less transparent than other contexts (such as negation) but more transparent than belief contexts. In honor of this fact, I will say that it is a translucent context—somewhat transparent, not completely opaque. The same holds for causal contexts: they are not fully transparent, but they are not as opaque as belief contexts either. For example, it can be true to say “The batsman is unconscious because the cricket ball hit him in the head”, but not true to say “The batsman is unconscious because the red ball hit him in the head”. The reason is that a cricket ball is always of a certain hardness and weight but a red ball isn’t—even if the cricket ball is in fact red. Being a cricket ball is causally relevant but being a red ball isn’t. This is even clearer if the context is explicitly explanatory: the explanation of the batsman’s unconsciousness is his being hit by a ball of a certain density and weight not the fact that the ball was red. We can substitute any description for the original description if it preserves reference to the causally relevant properties of the missile (“the heavy hard sphere travelling at 60 mph”), even if no one has any beliefs about these descriptions. The context is somewhat transparent, but not as transparent as a negation context—yet it falls short of the opacity of a belief context. It is transparent with respect to causally relevant explanatory properties, though not with respect to objects as such.

Belief contexts, as already implied, are opaquer than modal, causal, and explanatory contexts, because they bring in mental representation—ways of conceiving things, modes of presentation, concepts, perspectives. But they are not fully opaque, because they do allow some latitude for substitution—they don’t resist all substitutions. Thus, we may substitute synonymous terms within their scope: they are transparent with respect to sense. We could describe them as “semi-opaque” (or “semi-transparent”). Their substitutivity properties fall between full transparency and translucency, on the one hand, and a yet stricter kind of opacity, on the other. This third kind of opacity belongs to quotational contexts: they won’t even allow substitution of synonyms—you can only substitute terms that refer to the same terms (words, bits of language). These contexts may be called “hyper-opaque”, though simply calling them “opaque” would not be semantically amiss given that that word connotes an absolute condition. We have already concluded that belief contexts are not maximally opaque given their openness to synonyms—hence “semi-opaque”. So, we now have a four-way distinction: transparent, translucent, semi-opaque, and opaque (or hyper-opaque). It is a not a binary business, an all-or-nothing affair. Contexts are transparent with respect to this or that class of possible substitutions, opaque to one degree or another, not rigidly either one of the other. This is because words have a range of entities associated with them: references (objects in the world), aspects of objects (mind-independent properties), senses of words (ways of representing the world), and the words themselves (marks, sounds, states of the brain). The old extensional-intensional dichotomy is too simple.

Once we have made these distinctions in the case of singular terms, we can extend the apparatus to whole sentences. Some contexts (“and”, “not”) are truth-functional; some are fact-functional (“necessarily”, “because”, “explains”); some are sense-functional (“believes” and other propositional attitude verbs); and some are inscription-functional (direct speech, quotation). Being truth-functional has no special status on this way of looking at things; it is just an extreme type of truth-value dependence, i.e., any sub-sentence with the same truth-value can be substituted for the original without changing the truth-value of the whole. Some contexts permit a broader range of substitutions salva veritate than others: that is all. It isn’t some kind of special property of truth-values, as opposed to facts or senses, that renders them more respectable or pellucid. Nor are truth-functional contexts in any way superior because of their substitutional liberality: for all contexts have their own distinctive substitutional profile. It is true that the words “transparent” and “opaque” have different connotations, evaluatively speaking: transparency is “good”, opacity is “bad”—especially when it comes to discourse (not so much for clothes). But that is just an accident of terminology; it’s all the same merit-wise. We should be substitutivity egalitarians.

I have left till last a more startlingly unorthodox suggestion (the reader needed some softening up first). What about extending the concept of transparency beyond its normal bounds? To this end, I shall introduce the notion of extended extension: a given term extendedly refers not just to its normal referent but to a wider range of referents, e.g., any twin of the normal referent. This will allow a wide range of substitutions to retain truth-value, since most of what is true of an individual will also be true of its twin. In a universe consisting of Leibnizian pairs (indiscernible but numerally distinct individuals) everything true of one twin will be true of the other, so we can substitute one twin’s name for the other and not disrupt truth-value. The names will be transparent with respect to duplicates. This seems like a definable idea and it substantially expands the range of possible substitutions. Transparent contexts will be doubly transparent in this universe—hyper-transparent. We could also employ the distinction between objects and the matter that composes them, as in the statue and the piece of bronze that composes it. Again, nearly everything true of one will be true of the other, so substitution will be well-nigh universal. Thus, the simple dichotomy between transparent and opaque contexts yields to a more generous understanding of the semantic phenomena. This opens up new possibilities concerning formal models and interpretations, i.e., assignments of elements from a chosen domain. I rather warm to the idea of fourfold functions from domains onto the formalized language: objects, aspects, senses, and words themselves. Full substitutivity will hold with respect to each subdomain.[1]

[1] Informed readers will discern elements of Frege, Wittgenstein, Russell, Carnap, Quine, and others in this paper. In general, I am promoting greater inclusiveness in formal semantics. Many types of entity are relevant to semantic functioning. Jungle, not desert, landscapes.

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